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Russian Academy of Sciences \\
Siberian Division\\
Institute of Information Technologies and Applied Mathematics

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\LARGE
A.G.Myasnikov, V.N.Remeslennikov

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\Huge
Exponential groups

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Preprint $N$ 27
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Omsk 1996
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\noindent ISBN 5-7779-0027-5 \hfill \copyright A.~Myasnikov, \newline
\mbox{~} \hfill V~Remeslennikov, 1996


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\newtheorem{opr}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{pr}{Proposition}
\newtheorem{problem}{Problem}
\newtheorem{remark}{Remark}

{\Large
\centerline{\bf Exponential groups}
%\centerline{A.G.Myasnikov, V.N.Remeslennikov}
%\centerline{(New-York,  February 4, 1994)}
}

\begin{enumerate}

\item Introduction
\item Basic notions
\item Examples
\item A-commutator calculus.
\item Tensor completions
\item A-free groups and A-free products
\item Construction of centralizer extensions
\item CSA-groups
\item Faithful A-completion
\item Approximation theorems
\item Embeddings into formal power series.
\item Length functions.
\item Open problems
\item References

\end{enumerate}

\section {Introduction}

In this talk we observe some results which have been obtained by the athors
during the last two years in the theory of exponential groups. This theory
begins with  results of A.Mal'cev, P.Hall, R.Lyndon and G.Baumslag.
The  axiomatic notion of an exponential group was introduced by R.Lyndon (1960).
We add a new axiom to  Lyndon's definition and obtain a new notion of
an exponential group. The refined notion is more convenient because it
coincides exactly with the notion of a module over a ring in the
abelian case, whereas abelian exponential groups after Lyndon provide
a far wider class.

\newpage

\section{Basic notions}

Let $A$ be an arbitrary associative ring with identity and $G$ a group.
Fix an action of the ring $A$ on $G$, i.e. a map $G \times A \rightarrow
G$. The result of the action of $\alpha \in A$ on $g \in G$ is written
as $g^\alpha$. Consider the following axioms:

\begin{enumerate}
\item $g^1=g$, $g^0=1$,  $1^\alpha = 1$ ;
\item $g^{\alpha +\beta}=g^\alpha \cdot g^\beta, \ g^{\alpha \beta} =
(g^\alpha)^\beta$;
\item $(h^{-1}gh)^\alpha = h^{-1}g^\alpha h;$
\item $[g,h]=1 \Longrightarrow (gh)^\alpha = g^\alpha h^\alpha.$
\end{enumerate}

\begin{opr}
Groups $G$ with axioms 1)--2) are called $A$--groups, 1)--3)
$L_A$--groups, and 1)--4) $M_A$--groups. The corresponding classes
of groups are denoted by ${\cal R}_A, {\cal L}_A$ and ${\cal M}_A$.
\end{opr}

In our report we will consider only $M_A$--groups  and call them
simply  $A$--groups.\par

One can consider $A$--groups in two signatures: the pure group-theoretic
signature $\sigma = \langle \cdot, ^{-1}, 1 \rangle$ and the signature
of $A$--operator groups $\sigma_A = \sigma \cup \{f_\alpha (x) \mid \alpha
\in A \} $, where $f_\alpha(x)=x^\alpha$. Simple examples show that two
groups can be isomorphic as abstract groups, but not isomorphic as
$A$--groups. The class ${\cal L}_A$ is a variety and ${\cal M}_A$
is a quasi-variety in the signature
$\sigma_A$.\par
As usual, one can introduce the notions of $A$--torsion, finitely
$A$-generated group, $A$--homomorphism, $A$--isomorphism, and so on.

\section{Examples}

\begin{enumerate}
\item An arbitrary group is a ${\bf Z}$--group;
\item a divisible group from ${\cal L}_{\bf Q}$ is a ${\bf Q}$--group;
\item a group of period $m$ is a ${\bf Z}/m{\bf Z}$--group;
\item an arbitrary $A$--operator group from ${\cal L}_A$ with the
ring of operators $A$ is an $A$--group;
\item a module  over a ring $A$ is an abelian $A$--group;
\item free Lyndon $A$--groups are $A$--groups;
\item exponential nilpotent $A$--groups over a binomial ring $A$
introduced by P.Hall are $A$--groups;
\item an arbitrary pro-p group is a ${\bf Z}_p$--group over the
ring of p-adic integers ${\bf Z}_p$;
\item an arbitrary profinite group is a $\hat{{\bf Z}}$--group, where
$\hat{{\bf Z}}$ is the total completion of ${\bf Z}$ in the profinite
topology;
\item complex (real) unipotent groups are ${\bf C}$--$({\bf R}$--) groups.
\end{enumerate}

\section{A--commutator calculus}

The commutator collection process is well known in group theory. One
can define a generalization of this process in the class of
$A$--groups. A central notion here is the notion of an $\alpha$--commutator.

\begin{opr}
For $g,h \in G, \ \alpha \in A$ the element
$$
(g,h)_\alpha = h^{-\alpha}g^{-\alpha}(gh)^\alpha
$$
is called an $\alpha$--commutator of the elements $g$ and $h$.
\end{opr}

It is clear that $(gh)^\alpha = g^\alpha h^\alpha (g,h)_\alpha;$
$ G \in {\cal M}_A \Longleftrightarrow ([g,h]=1 \longrightarrow
\forall \alpha (g,h)_\alpha = 1)$.

\begin{opr}
Let $G \in {\cal L}_A$. Then a normal $A$--subgroup $H \lhd G$ is called
an ${\cal M}_A$-- ideal if for any $g,h \in G, \ \alpha \in A$ we have
$$
[g,h] \in H \Longrightarrow (g,h)_\alpha \in H
$$
\end{opr}

\begin{pr}
Let $G \in {\cal L}_A$. Then
\begin{enumerate}
\item if $\varphi : G \rightarrow H$ is an $A$--homomorphism of groups
from ${\cal M}_A$ then $ker \varphi$ is an ${\cal M}_A$ -- ideal in $G$;
\item if $H$ is an ${\cal M}_A$--ideal in $G$ then $G/H \in {\cal M}_A$.
\end{enumerate}
\end{pr}

\begin{opr}
An ${\cal M}_A$--ideal $(G,G)_A=id_{\cal M}([G,G])$ generated by the
ordinary commutator subgroup
of an $A$--group $G$ is called the $A$--commutator subgroup of $G$.
\end{opr}

\begin{pr}
\begin{enumerate}
\item $(G,G)_A=gr_A\langle (g,h)_\alpha \mid g,h \in G, \alpha \in A \rangle$
\item $(G,G)_A$ is the least ${\cal M}_A$--ideal in $G$ with an abelian
quotient $G/(G,G)_A$.
\end{enumerate}
\end{pr}

Axioms 1)--4) give us some identities on $\alpha$--commutators which
allow one to perform a collecting process.

\section{Tensor completion}

The basic operation in the class of $A$--groups is the operation of
tensor completion.
\begin{opr}
Let $G$ be an $A$--group and $\mu : A \rightarrow B$ a homomorphism of
rings. Then a $B$--group $G^{B, \mu}$ is called  a tensor $B$--completion
ofthe group $G$ if $G^{B, \mu}$ satisfies the following universal
property:
\begin{enumerate}
\item there exists an $A$--homomorphism $i: G \rightarrow G^{B, \mu}$
such that $i(G)$ $B$--generates $G_{B, \mu}$, i.e.
$$
\langle i(G) \rangle_B = G^{B, \mu}
$$
\item for any $B$--group $H$ and a homomorphism $\varphi: G \rightarrow H$
compatible with $\mu$ (i.e. $(g^\alpha)^\phi = (g^\phi)^{\mu \alpha})$
there exists a $B$--homomorphism $\psi: G^{B, \mu} \rightarrow H$
such that the following diagram commutes:

\medskip

\begin{center}

\begin{picture}(100,100)(0,0)
\put(0,100){$G$}
\put(100,100){$G^{B, \mu}$}
\put(0,0){$H$}
\put(15,103){\vector(1,0){80}}
\put(5,93){\vector(0,-1){78}}
\put(95,95){\vector(-1,-1){80}}
\put(-10,50){$\phi$}
\put(55,45){$\psi$}
\put(50,108){$i$}
\end{picture}

\end{center}

\medskip

\end{enumerate}
\end{opr}
If $G$ is an abelian $A$--group, then the group $G^{B, \mu}$ is also
abelian, i.e. it is a $B$--module. $G^{B, \mu}$ satisfies the same
universal property as the tensor product $G \otimes_A B$ of the
$A$--module $A$ and the ring $B$. Therefore
$$
G^{B, \mu} \simeq G \otimes_A B
$$
\begin{theorem}
Let $G$ be an $A$--group and $\mu$ a homomorphism of rings. Then there
exists a tensor completion $G^{B, \mu}$.
\end{theorem}

\begin{theorem}
Let $G$ be an $A$--group and $\mu: A \rightarrow B$ a homomorphism
of the rings $A$ and $B$. Then the tensor completion is unique up to a
$B$--homomorphism.
\end{theorem}

Let $\mu $ be a homomorphism of rings $A$ and $B$. On the basis
of this homomorphism the construction
of tensor completion allows one to construct a functor $\theta^{B, \mu}$
:${\cal M}_A \rightarrow {\cal M}_B$.

\begin{pr}
Let $\mu: A \rightarrow B$ be a homomorphism of rings. Then $\theta^{B, \mu}$
is a covariant functor from the category ${\cal M}_A $ to the category
${\cal M}_B$.
\end{pr}

With the help of this proposition theorems on the construction of tensor
completion can be reduced to the following two cases:
\begin{enumerate}
\item[a)] $\mu$ is an epimorphism of rings;
\item[b)] $\mu$ is a monomorphism of rings.
\end{enumerate}

One can also prove that the operation of tensor completion
commutes with the operations of direct product and direct limit.

\section{A--free groups and A--free products}

\begin{opr}
An $A$-group $F_A(X)$ with the set of $A$-generators $X$ is
said to be a free $A$-group with base $X$, if for every
$A$-group $G$ an arbitrary mapping $\phi_0:X \mapsto G$ can
be extended to an $A$-homomorphism $\phi: F_A(X) \mapsto G$.
The set $X$ is called the set of free $A$-generators of
$F_A(X)$. The cardinality of $X$ is called the rank of
the group $F_A(X)$.
\end{opr}

\begin{theorem}
For every $X$ and $A$ there exists a free $A$-group $F_A(X)$;
moreover it is unique up to isomorphism and $F_A(X) \simeq (F(X))^A$.
\end{theorem}

Let us introduce the construction of a free product in the
category of $A$-groups.

\begin{opr}
Let $G_i, \ i \in I$, be $A$-groups. An $A$-group
$\ast_A G_i$ is called the free product in the category
${\cal M}_A$, if the $A$-homomorphisms $\phi_i:G_i \mapsto
\ast_AG_i$ are such that for every family of $A$-homomorphisms
$\psi_i:G_i \mapsto H$, where $H$ is an arbitrary $A$-group,
there exists an $A$-homomorphism $\psi: \ast_AG_i \mapsto H$
such that the following diagram commutes for all $i \in I$

\medskip

\begin{center}

\begin{picture}(100,100)(0,0)
\put(0,100){$G_i$}
\put(100,100){$\ast_AG_i$}
\put(0,0){$H$}
\put(15,103){\vector(1,0){80}}
\put(5,93){\vector(0,-1){78}}
\put(95,95){\vector(-1,-1){80}}
\put(-10,50){$\phi_i$}
\put(55,45){$\psi$}
\end{picture}

\end{center}

\medskip

and $\ast_AG_i$ is $A$-generated by the set $\{\phi_i(g_i) \mid
g_i \in G_i, \ i \in I\}.$
\end{opr}

\par
It follows from category theory that the group $\ast_AG_i$
is defined uniqely up to an $A$-isomorphism.

\begin{theorem}
Let $A$ be a ring containing the ring of integers {\bf Z} as a
subring, and $\{G_i \ \mid \ i \in I\}$ is a set of $A$-groups.
Then $\ast_AG_i \ \simeq \ (\ast G_I)^A$.
\end{theorem}


\section{Construction of centralizer extensions}

For an explicit construction of tensor completion we will use the
construction of free extensions of centralizers for a given group $G$.
Let us describe this construction. \par
Let $G$ be a group. The centralizer of an element $v \in G$ is
denoted by $C_G(v)$.

\begin{opr}
Let $G$ be a group, $C_G(v)=C$ the centralizer of an element
$v$ from $G$, and $\phi :C\longrightarrow H$  a monomorphism
of groups such that $\phi (v)\in Z(H)$. Then the group
$$
G(C,H)=\langle G\ast H \mid C=C^\phi \rangle
$$
is called a free extension of the centralizer $C_G(v)$ by the group
$H$ with respect to $\phi$.
\end{opr}
\par
Some particular cases of this construction are of special interest.
We will distinct these kinds of extensions of centralizers with
respect to the type of the embedding $\phi:C \longrightarrow H$:
\begin{enumerate}
\item {\it isolated} iff $C^\phi$ is an isolated subgroup of $H$;
\item {\it splitting} iff $H=C^\phi\cdot B$ for some subgroup $B$;
\item {\it direct} iff $H=C^\phi \times B$.\par
Also, we will differ extensions by the type of the group $H$.
For example, an extension of a centralizer is
\item {\it abelian} iff $H$ is an abelian group.
\end{enumerate}

Now we will describe a construction which allows one to extend a
set of centralizers at once.
\par
\begin{opr}
Let ${\cal C}=\{C_i = C_G(v_i) \mid i \in I \}$ be a set
of centralizers in the group $G$. Suppose $\phi_i: C_i
\longrightarrow H_i$ is an embedding of $C_i$ into $H_i$
such that $\phi_i(v) \in Z(H_i), \ i \in I$. We can form a
graph of groups


\medskip

\begin{center}

\begin{picture}(100,100)(0,0)
\put(0,50){$G$}
\put(100,100){$H_i$}
\put(100,50){$H_j$}

\put(103,75){\vdots}
\put(103,25){\vdots}
\put(15,57){\line(2,1){75}}
\put(15,53){\line(1,0){75}}
\put(50,85){$C_i$}
\put(50,60){$C_j$}
\end{picture}

\end{center}

\medskip

The fundamental group of this graph is called a tree extension of
centralizers from ${\cal C}$ and is denoted by $G({\cal C, H}, \Phi)$,
where ${\cal H}=\{H_i, \ i \in I \},\ \Phi=\{ \phi_i, \ i \in I \}$.
\end{opr}
\par
Again, we will consider central, isolated, splitting, direct and
abelian extensions corresponding to the type of monomorphisms
from $\Phi$.\par
Finally, we introduce one more construction: the iterated tree
extension of centralizers.
\par
Let $G$ be a group, ${\cal C}=\{C_G(v_i) \mid i \in I \}$ a set
of centralizers in $G$, and $\Phi = \{\phi_i: C_G(v_i) \rightarrow
H_i \}$ a set of embeddings of these centralizers. Then one can
construct the group $G_1=G({\cal C, H}, \Phi)$ and consider again
the set of centralizers ${\cal C}_1$ in $G_1$, the set of embeddings
$\Phi_1$, and the corresponding set of groups ${\cal H}_1$. As above,
the group $G_2 = G_1({\cal C}_1, {\cal H}_1, \Phi_1)$ can be
constructed. We can repeat this process using induction on ordinals:
if a group $G_\alpha$ has been constructed for some ordinal
$\alpha$ and the set of centralizers ${\cal C}_\alpha$ in $G_\alpha$,
the set of groups ${\cal H}_\alpha$ and the corresponding set of
embeddings $\Phi_\alpha$ have been chosen, then $G_{\alpha+1} =
G_\alpha({\cal C}_\alpha, {\cal H}_\alpha, \Phi_\alpha)$.
For a limit ordinal $\lambda$
$$
G_\lambda = \lim_{\stackrel{\longrightarrow}{\alpha <\lambda}} G_\alpha
$$
Thus,
for any ordinal $\delta$, starting from a given group $G$, we can obtain
the group $G_\delta$ by iterating the process above.

\begin{opr}
The group $G_\delta$ is called an iterated tree extension of centralizers
up to level $\delta$.
\end{opr}

\section{CSA--groups}

Here we will introduce a class of groups which will play a
fundamental role in our investigation of $A$-free groups, tensor
completions and groups with length functions. This class is quite
wide; for example, it contains all torsion-free hyperbolic groups
and all groups acting freely on $\Lambda$--trees. A very important
feature of this class is its good behavior under abelian extensions
of centralizers.
\begin{opr}
A subgroup $H$ of a group $G$ is called {\it conjugately separated} if
$H \cap H^x = 1$ for any $x \in G \setminus H$.
\end{opr}

\begin{opr}
A group G is called a CSA-group if all its maximal abelian subgroups
are conjugately separated.
\end{opr}

\par
The following proposition gives us some intuition about CSA-groups.

\begin{pr}
\label{3.1}
Let $G$ be a CSA-group. Then the following statements are true:
\begin{enumerate}
\item (SA--property) if $M_1$ and $M_2$ are distinct maximal abelian
subgroups of $G$, then $M_1 \cap M_2 = 1$;
\item if $x,y \in M$, where $M$ is a maximal abelian subgroup of $G$,
then $x$ is conjugate to $y$ in $G$ only if $x=y$;
\item commutation is an equivalence relation on the set
$G \setminus \{1\}$ of all nontrivial elements of $G$ (this condition
is equivalent to the following: any centralizer of a nontrivial
element og $G$ is abelian);
\item any maximal abelian subgroup of $G$ coincides with its centralizer
and normalizer.
\end{enumerate}
\end{pr}
\begin{pr}
\par
\begin{enumerate}
\item The class of groups CSA is finitely universally axiomatizible.
\item The class CSA is closed under ultraproducts.
\end{enumerate}
\end{pr}

\begin{pr}
\label{3.6}
Let $G$ be a group such that
\begin{enumerate}
\item[a)] $G$ is torsion-free;
\item[b)] any centralizer of a nontrivial elements is cyclic.
\end{enumerate}
Then $G$ is a CSA-group.
\end{pr}

\begin{pr}
\label{3.7}
\par\medskip
\begin{enumerate}
\item Any torsion-free hyperbolic group is a CSA-group.
\item Any group acting freely on a $\Lambda$-tree is a CSA-group.
\item Any one--relator group without Karrass--Baumslag subgroups is a
CSA--group.
\end{enumerate}
\end{pr}

\begin{opr}
The subclass CSA$^\ast$ of CSA consists of all groups without elements
of order two.
\end{opr}
\begin{remark}
Groups from proposition~\ref{3.7} are CSA$^\ast$--groups.
\end{remark}
\begin{pr}
The class CSA$^\ast$ is closed under free products.
\end{pr}

The following theorems play an important role.

\begin{theorem}
The class CSA$^\ast$ is closed under abelian extensions of centralizers.
\end{theorem}

\begin{theorem}
The class CSA$^\ast$ is closed under tree abelian extensions of
cenralizers and iterated tree abelian extensions of centralizers.
\end{theorem}

\section{Faithful A--completion}

It is convenient to construct the tensor completion of a given group
step by step, defining powers gradually. This involves the notion of
a partial $A$--group.

\begin{opr}
We call a group $G$ a partial $A$--group if the action of the ring $A$
is defined for some pairs $(g, \alpha)$, but is not necessarily
defined for all pairs; and if both parts of an equation in the axioms
1)--4) are defined, then the corresponding axiom holds.
\end{opr}

Let $G$ be a partial $A$--group. In particular, if $A_0$ ia a
subring of $A$, then any $A_0$--group is also a partial
$A$--group. So consideration of partial $A$--groups
covers the case of tensor $A$--completion of $A_0$--groups
from the paper [ ].

\begin{opr}
A partial $A$--group $G$ is called faithful (over $A$),
if the canonical homomorphism of partial $A$--groups
$\lambda: G \longrightarrow G^A$ is injective.
\end{opr}

Now we'll describe the explicit construction of a tensor completion
of an arbitrary CSA$^\ast$--group $G$ using extensions of centralizers.
In fact, $G$ appears as an iterated tree extension of centralizers in $G$
for a very wide class of rings $A$. In particular, $G^A$ is again a
CSA$^\ast$--group, and we obtain a lot of information about it.

\begin{opr}
A group $G \in {\cal P}_A$ is called a PCSA--group if it satisfies
the following conditions for any maximal abelian subgroup $M$:
\begin{enumerate}
\item $M$ is conjugately separated;
\item the canonical homomorphism of partial $A$--groups
$i: M \longrightarrow M \otimes A$ is injective.
\end{enumerate}
If $G$ is a PCSA--group and has no elements of order 2, we will
call it a PCSA$^\ast$--group.
\end{opr}

\begin{theorem}
Let $G$ be a PCSA$^\ast$--group, $A$ a ring, and let $A^+$ have no
elements of order 2. Then the tensor completion of $G$ by $A$ is an
iterated tensor extension of centralizers of $G$ by the ring $A$.
Moreover, $G^A$ is again a CSA$^\ast$--group.
\end{theorem}

\begin{theorem}
Let $G$ be a torsion--free CSA--group and $A$ a ring with a torsion--free
additive group $A^+$. Then $G^A$ is a torsion--free CSA--group, and the
canonical map $i: G \longrightarrow G^A$ is injective. In particular,
this is true for a hyperbolic torsion--free group $G$, or for a group
$G$ acting freely on a $\Lambda$--tree.
\end{theorem}


We have constructed the group $G^A$ with the help of the
construction of free extensions of centralizers, which
doesn't use the multiplicative structure of the ring. Therefore
if $G \in CSA^{\ast}$, and an abelian group $A$ (not necessarily
the additive group of a ring) contains the images of all
centralizers which were used in this construction, then we can
consider the tensor completion $G^A$ of $G$ with respect to
any abelian group $A$ of this type. If $B$ is a subgroup of $A$
containing the image of ${\bf Z}$, we can naturally consider
$G^B$ as a subgroup of $G^A$.

\section{Approximation theorems}
Throughout this section, $F$ stands for a non-abelian free group.

\begin{opr}
Let $G$ and $H$ be groups. Then $G$ is $\omega$-residually $H$,
if for any finite set of nontrivial elements from $G$ there exists
a homomorphism from $G$ into $H$ such that the images of the
chosen elements are nontrivial in $H$.\par
If in the situation above the statement is true only for
one-element sets, then we have the ordinary notion of residually
$H$--groups.
\end{opr}

We will formulate approximation theorems in two signatures:
the purely group signature $\sigma$ and the signature
$\sigma_A$. We will consider only the case where the additive
group $A^+$ is torsion--free. Due to the type of the embedding
of ${\bf Z}$ into $A^+$, we will distinct two cases:
\begin{enumerate}
\item[a)] (Lyndon's case) ${\bf Z}$ is a pure subgroup in $A^+$;
\item[b)] (Baumslag's case) ${\bf Z}$ is not a pure subgroup in $A^+$.
\end{enumerate}


{\bf Case 1}. Group signature.\par \medskip
\begin{enumerate}
\item[a)] {\bf Lyndon's case.}
\begin{theorem}[Local approximation theorem]
Let $F$ be a free group. Let ${\bf Z}$ be a pure subgroup in $A^+$,
and $B$ a finitely
generated subgroup of $A$ containing ${\bf Z}$.  Then the group
$F^B$ is $\omega$-residually free.
\end{theorem}
\begin{theorem}[Global approximation theorem]
Let $F$ be a free group. Let ${\bf Z}$ be a pure subgroup in $A^+$,
and $A^+$  $\omega$-residually ${\bf Z}$ (or, equivalently,
residually ${\bf Z}$). Then the group $F^A$ is $\omega$--residually free.
\end{theorem}

\item[b)] {\bf Baumslag's case.}
\par \medskip
Let $A_{\bf Q}$ be $A \otimes {\bf Q}$, and let $F$ be a free group.
Then $F^A$ is a subgroup of $F^{A_{\bf Q}}$.

\begin{theorem}[Global approximation theorem]
Let $F$ be a free group. \\ Then the group $F^{A_{\bf Q}}$ is
$\omega$-residually $F^{\bf Q}$.
\end{theorem}

\end{enumerate}

{\bf Case 2} Signature $\sigma_A$. \par \medskip

The notion of $\omega$--residually (residually) can be naturally
defined for the category of $A$--groups. The corresponding homomorphism
in the definition above should be an $A$--homomorphism.
\begin{enumerate}
\item[a)] {\bf Lyndon's case.}
\begin{theorem}[Global approximation theorem]
Let the ring $A$ be residually ($\omega$--residually) ${\bf Z}$.
Then the $A$--group $F^A$ is $\omega$--residually free in the category
of $A$--groups.
\end{theorem}
\begin{theorem}[Local approximation theorem]
Let the ring $A$ be locally \linebreak 
residually (locally $\omega$--residually)
${\bf Z}$, and $B$ a finitely generated subring of $A$.
Then for $F^B$  the global approximation theorem with respect to $B$
is true.
\end{theorem}

\item[b)] {\bf Baumslag's case.}\par \medskip
Let $A$ be a ring, and {\bf Z} not a pure subgroup in $A^+$.
Denote by $A_{\bf Q}$ the ring $A \otimes {\bf Q}$. The latter
ring is a ${\bf Q}$--algebra.
\begin{theorem}[Global approximation theorem]
Let the ${\bf Q}$--algebra $A_{\bf Q}$ be residually ($\omega$--residually)
${\bf Q}$. Then $F^{A_{\bf Q}}$ is residually ($\omega$--residually)
$F^{\bf Q}$ in the category of $A_{\bf Q}$--groups.
\end{theorem}
\end{enumerate}

\begin{pr}
Let $G$ be  residually $F$. Then :
\begin{enumerate}
\item $G$ is residually nilpotent torsion--free;
\item $G$ is a subgroup of $SL(2,K)$ for some field $K$.
\end{enumerate}
\end{pr}

{\bf Corollaries to approximation theorems}
\begin{enumerate}
\item Let $F^B$ be as in the local approximation theorem (Lyndon's case).
Then:
\begin{enumerate}
\item[a)] $F^B$ is residually nilpotent torsion--free;
\item[b)] $F^B$ can be embedded into $SL(2,K)$ for some field $K$;
\item[c)] $F^B$ is an E--free group.
\end{enumerate}
\item Let $A$ be residually ${\bf Z}$ and ${\bf Z}$ a pure subgroup
of $A$ (global Lyndon's case). Then $F^A$ satisfies items a)-c) above.
\item Let the ring $A$ be residually ${\bf Z}$. Then $F^A$ is a residually
nilpotent torsion--free $A$--group.
\end{enumerate}



\section{Embeddings into formal power series}
Let $A$ be a binomial integral domain, $R=A[[X]]$ the ring of formal power
series, $X=\{ x_1, \ldots, x_n \}$, and $\Delta$ the fundamental
ideal of $R$. Then $1+\Delta$ is
a multiplicative subgroup of $R$. One can define an action of $A$
on $1+\Delta$ in the following way:
$$
(1+f)^\alpha = 1+\alpha f + (^\alpha_2)f^2 + \ldots
$$
This turns $1+\Delta$ into an $A$--group. Moreover, $1+\Delta^n$ is
an ${\cal M}_A$--ideal in $1+\Delta$, and $(1+\Delta)/(1+\Delta^n)$
is a torsion--free $A$--nilpotent group in the sense of P.Hall.
It is a well--known result of Magnus that the map $x_i
\mapsto 1-x_i$ can be extended
to an embedding of the free group $F(X)$ into the multiplicative
group of the ring   ${\bf Z}[[X]]$. One can generalize this result to
the global Lyndon's case.
\begin{theorem}[Global Lyndon's case]
Let $A$ be a binomial ring which is residually ${\bf Z}$.
Then the map $x_i \mapsto  1-x_i$ can be extended to an
$A$--embedding of $F(X)^A$ into $1+\Delta$.
\end{theorem}

In the local Lyndon's case the situation is a little bit worse.
In general, we can embed $F(X)^A$ into the ring of formal power
series $A[T][[X]]$, where $A[T]$ is a polynomial ring with
a sufficiently large set of indeterminates $T$.
\begin{theorem}[Local Lyndon's Case]
Let the subgroup ${\bf Z}$, which is generated by the unit in $A$,
be a pure subgroup of $A^+$. Then the map $x_i \mapsto 1-x_i$
can be extended to an embedding of $F(X)^A$ into $A[T][[X]]$,
where the cardinality of $T$ is equal to max$\{$card($F(X)$), card($A)\}$.
\end{theorem}



\section{Length functions}

Let $G$ be a group and $\Lambda$ an ordered abelian group. Then a function
$l: G \rightarrow \Lambda$ is called a {\it length function} on the group $G$
with values in the group $\Lambda$ if the following condition hold:

$
1) \forall x \; l(x) \geq 0,\ l(1)=0;
$

$
2) \forall x \; l(x)=l(x^{-1});
$

$
3) \forall x \forall y \forall z \;
    (d(x,y)>d(x,z) \Rightarrow d(x,y)=d(y,z)),
$

$
\ \ \ where \ d(x,y)=\frac{1}{2}(l(x)+l(y)-l(x^{-1}y)).
$

To define $d(x,y)$ correctly one can assume that $\Lambda$ is canonically
embeded in its divisible completion $\Lambda \otimes {\bf Q}$, which is
an ordered abelian group and the order extends the order on $G$. From
other hand one can formulate the axiom 3) using the function
$d_0(x,y)=2d(x,y)$ instead of $d(x,y)$.

Moreover, the length function $l:G \rightarrow \Lambda$ is called
{\it free length function}, if $l$ satisfies the following axiom

$
4) \forall x \; (x \neq 1 \rightarrow l(x^2) > l(x).
$

From now on we fix a free length function $l: G \rightarrow \Lambda$.
For elements $x_1, \ldots, x_n \in G$ the notation
$x=x_1 \circ \ldots \circ x_n$ means that $x=x_1 \ldots x_n$ and
$l(x) = l(x_1)+ \ldots +l(x_n)$.

\begin{opr}
The length function $l: G \rightarrow \Lambda$ is called {\sf regular}
iff the following axiom holds:

$
5) \forall x \forall y \exists u \exists x_1 \exists y_1 \;
(x=u \circ x_1 \& y=u \circ y_1 \& l(u)=d(x,y))
$

\end{opr}

Now we start to discuss the situation when each element of $G$
can be decomposed into a product of further indecomposable elements (one
can keeps in mind a free froup with some fixed base).

\begin{opr}
An element $a \in G$ is called {\sf an atom} (with respect to the length
function $l: G \rightarrow \Lambda$) if the following conditions holds:

$
6) \forall x \forall y  \;(a=x \circ y \rightarrow l(x)=0 \vee l(y)=0).
$
\end{opr}

Suppose that an ordered abelian group $\Lambda$ has a series of
convex subgroups:
$$
0=\Lambda_0<\Lambda_1< \ldots <\Lambda_i< \ldots
$$

$I$ is a well-ordered set. The preimage $l^{-1}(\Lambda_i)=G_i$ is
a subgroup in $G$, so the series in $\Lambda$ induces a series of
subgroups in $G$:
$$
1 \leq G_0 \leq G_1 \leq \ldots \leq G_i \leq \ldots, \ i \in I
$$

Let us denote the restriction of on $\Lambda_i$ by $l_i$. Then
$l_i: G_i \rightarrow \Lambda_i$ is a length function on $G_i$.
The order on $\Lambda_i$ induces an order on
$\overline{\Lambda_i} = \Lambda_{i+1} / \Lambda_i$ it allows one to
consider the induced length function
$\overline{l_i}: G_i \rightarrow \overline{\Lambda_i}$

\begin{opr}
Let two elements from $G_i$ are called {\sf equivalent} ($g_1 \sim g_2$)
if the following conditions holds:
$$
\exists x \exists y \in G_{i-1} \; g_1=xg_2y
$$
\end{opr}


\begin{opr}
An element $b \in G_i$ is called {\sf an i-block (or a generalized i-atom)}
if the following conditions holds:

$\forall x \forall y \; (b=x \circ y \rightarrow x \in G_{i-1} \vee
y \in G_{i-1} \vee \exists x_1 \;(x_1\sim x) \;\exists y_1 \;(y_1 \sim y)$
s.t. $[x_1,y_1]=1)$.
\end{opr}


\begin{opr}
The length function $l: G \rightarrow \Lambda$ is called {\sf atomness
(blockness)} if it satisfies the following axiom:

$
8) \forall x \; ( \bigvee_{i=0}^{\infty} \exists y_1, \ldots y_i \;
(x=y_1 \circ \ldots \circ y_i \& y_1, \ldots y_i \ are \ atoms \ (blocks))).
$
\end{opr}

\begin{opr}
A length function $l: G \rightarrow \Lambda$ is called {\sf step-like}
if for any $i \in I$ the length function
$\overline{l_i}: G_i \rightarrow \overline{\Lambda_i}$ is atomness
(or blockness).
\end{opr}



Let  $l: G \rightarrow \Lambda$ be a length function on a group $G$ and
$V=\{ v_i \mid i \in I \}$ a set of elements from $G$. We will say that
$l$ satisfies {\it stabilizing condition} (S) with respect to $V$ if for any
elements $v, w \in V \cup \{1\} \; g \in G,$ with the property -- if $v=w$, then
$[g,v] \neq 1$ --, there exists a
natural number $t=t(v,g,w)$ such that $\forall n,m \geq t$
$$
v^n g w^m = v^{n-t} \circ v^t g w^t \circ w^{m-t}.
$$

Let $A$ be an associative ring with identity 1, its additive group $A^+$ is
torsion-free and the additive subgroup $<1>$ is a pure subgroup of $A^+$.
Our goal is to construct a Lyndon free step-like length function for a free
exponential group $F^A$.
The length function on $F^A$ is constructed by induction. The
following technical theorem provides us with the induction step from $G_i$
to $G_{i+1}$ (see free exponential groups).

\begin{theorem}  \label{tt}

Let $G$ be a group with a length function  $l: G \rightarrow \Lambda_0$
which satisfies conditions:

1. $l$ is a free regular step-like function;

2. stabilizing condition (S) for $V({\bf C}) = \{v_i \mid i \in I \}$. \newline
Assume ${\bf C} = \{�_i \mid i \in I \}$ is a set of centralizers in $G$
such that:

3. $C_i$ is an infinite cyclic group $\bf Z$, $C_i=<v_i>$ and
$l(v_i^n) = \mid n \mid l(v_i),\ n \in \bf Z$

4. $C_i$ is not conjugated to $G_j$ for $i \neq j$. \newline
Suppose that $G^*=G({\bf C,Z} \oplus A)$ is obtained from $G$ by tree
extension of centralizers from $\bf C$ by the group ${\bf Z} \oplus A$
under amalgamation $C_i$ and $Z$. \newline
Fix any order on the $A= \Lambda$ and introduce a right lexicographic
order on $\Lambda_0 \oplus \Lambda$.

Then the length function  $l: G \rightarrow \Lambda_0$ can be extended to
a length function  $l^*: G^* \rightarrow \Lambda_0 \oplus \Lambda$ for
which conditions 1)-2) hold under the chain of convex subgroups
$0 < \Lambda_0 < \Lambda_0 \oplus \Lambda$.

\end{theorem}

With the help of  theorem \ref{tt} one can prove

\begin{theorem}

Let $F^A$ be a  free  exponential group of Nielsen type.
Then there exists  a free regular step-like length function on the group
$F^A$.

\end{theorem}







\section{Open problems}

As was mentioned above,  in Lyndon's case the free group $F^A$
is residually torsion free nilpotent and linear. Naturally, the
following problems arise:
\begin{problem}[Baumslag] Let $A$ be a ring and $A^+$ torsion--free.
Then the group $F^A$ is residually torsion--free nilpotent.
\end{problem}

\begin{problem} Let $A$ be a ring and $A^+$ torsion--free.
Then the group $F^A$ is linear.
\end{problem}

\begin{problem}
Let $A$ be a ring and $A^+$ torsion--free. Then the universal theory
of the group $F^A$ is decidable.
\end{problem}

\begin{problem}
Generalize the notions of free constructions and hyperbolic groups
to the category
of $A$--groups.
\end{problem}

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\end{document}

